Building a Better Bracket, Part II

Yesterday I discussed a technique for squeezing a round out of the lower bracket in a double-elimination tournament by shifting some of the drops into early rounds. Now it’s time to discuss the other merits of that technique.

Here’s the bottom line: The round compression technique doesn’t just save a round so that the tournament runs faster, but it also improves overall fairness, and reduces the number of repeated pairings. It’s not just a plausible way to save some time in a round-intensive tournament, but it should probably be the technique of choice for almost any double-elimination tournament, whether or not there’s any benefit from running fewer rounds.

This is a rather surprising finding. I remember thinking, when I first learned of the technique, that it might well, in some contexts, be a sensible way to save time, but that it would turn out to have, upon close examination, some disadvantages that would militate against its use in any context in which the number of rounds is less than critical. So, let me make the case for the technique in terms of fairness and of the other goals of tournament design.

There are four design goals: fairness, efficiency, participation, and spectator appeal. But perhaps this really ought to be six, because there are three aspects of fairness, and they often conflict with one another in a particular context.

Begin with efficiency. The main purpose, as far as I saw it, was to save rounds when rounds counted. If the number of rounds is not important, this aspect is probably a wash – it’s no more difficult, and no easier, to run a round-compressed tournament.

In the example tournament, the standard brackets take eight rounds to resolve, the revised brackets only seven. Advantage to the shifted bracket.

Participation also weighs in favor of the shift. There will be the same number of games played, but somewhat fewer of them will be repeated pairings. As suggest in measuring participation, most players will feel they get more benefit when they play a greater variety of opponents.

From the simulation, with the standard bracket, there are 1.4 expected rematches. From the shifted bracket, 1.2 – advantage to the shift. It’s not an enormous difference, but in my experience most players really dislike rematches, so it’s worth something to eliminate a few of them.

Spectator appeal is probably a wash. It might be argued that spectators will be confused by the shift, but that is really a fairness (A) argument, which we’ll discuss presently. Otherwise, it’s hard to see how the bracket shift has any effect one way or the other on the watchability of the matches. They all remain the same sort of do-or-die elimination matches.

In some contexts, the shift may make slightly better watching because it slightly increases the chances for the better players to do well in the tournament. But it’s also true that in some cases some cases organizers would have an incentive not to save a round so that there would be another time slot in which to sell tickets or broadcast rights.

On to fairness. I’ll discuss the three sorts of fairness in reverse order.

Fairness (C) (rewarding superior play) is the only form of fairness for which I have a reasonably sophisticated numerical measure. The shifted bracket shows a modest advantage, 10.97 to 10.86. If your intuition was, as mine was, that there is something about the shifted bracket that isn’t quite right, the fairness calculation should have been able to show where the problem is. But the opposite is true. Tomorrow, I’ll examine the analyzed brackets in some detail to show why the shift not only doesn’t hurt, but helps a little.

Fairness (B) (equal chances) is reasonably implicated whenever the bracket is not uniform – where, as with the shifted bracket, some parts of the bracket face greater challenges than others. A close inspection of the analysis reveals that there are some such effects, but they are tiny. So tiny that I felt I had to run 100,000,000 simulations to be sure those effects were not random variations, and even so, they’re hard to spot. There is, perhaps, a fairness (B) objection to the shifted bracket, but the actual effect is negligible.

Finally, fairness (A) (settled expectations) resists definitive analysis because different constituencies will have different expectations about how tournaments should be run. My own first reaction to the shifted bracket was strongly negative, based on a theory I’d now characterized as a fairness (A) complaint. But I’ve now decided I was wrong.

In the next post, I’ll try to show how a close inspection of the analyzed brackets sheds light on all three aspects of fairness.